Small Neon Transformer??

I recently got 2 small transformers which are similar except on 2 points.  
Each is about the size of my fist and has 2 winding sections.  The secondary
section appears about twice the size of the primary.  Both sections appear
to be wound on a single long narrow stack of laminations which runs parallel
to the length of the transformer, and "in the middle" of the transformer.
Primary resistance readings are about 25 Ohms.

Secondary resistance reading is about 8K Ohms.  Does this seem normal??

Secondary resistance reading is about 50 Ohms.  Does this seem normal??

Has a third/middle wire lead on the primary and also on the secondary, and
these "extra" leads seem to be connected to each other, but not to the
primary or secondary windings.  What's the purpose of these "extra" leads?

At least one of these "extra" leads was originally connected to a 300V/2uF
cap; could this be for power factor correction?

Could these be 3KV neon transformers?  If so, what would a typical current
rating be for them?

Has anyone tried building a "mini" Tesla coil from such a transformer?

Thanks for any info!